A Survey of Central Limit Theorems in Number Theory

Abstract

In 1997, J-P Serre provided general principles for studying µ-equidistributed sequences in compact subsets of the real numbers. As an application, Serre obtained the asymptotic distribution of the eigenvalues of the Hecke operator T_p acting on spaces of modular cusp forms, as well as the eigenvalues of Hecke-type operators acting on families of regular graphs. In 2009, R. Murty and K. Sinha derived general principles for obtaining upper bounds in the discrepancies of equidistributed sequences. These principles can be viewed as “effective” versions of the Wiener-Schoenberg criterion, which generalises Weyl’s criterion for uniformly distributed sequences to µ-equidistributed sequences. In particular, this provides more information on the rate of convergence in the equidistribution of the families considered by Serre. This further leads us to study general principles about the fluctuations in the discrepancies of sequences picked uniformly at random from appropriate families. In analogy with central limit theorems, this thesis addresses under what conditions the distribution of these fluctuations matches the normal distribution.